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Fourier Analysis is the decomposition of any square-integrable functions into an infinite series of trigonometric functions. Here we show that the trigonometric polynomial functions are dense in the space of periodic continuous functions, and thus can be used as good approximations. This implies...

From Laplace's Calculus of Probability (Lecture 3), a finite difference equation explains probabilistically how a game evolves over discrete time, defining thus a stochastic process and specifically a Markov Chain where there is finite recursion or limited historical dependence. While one can...

We consider thus the functional notion of communication within the theory of dynamic systems, where the finite difference equation, or flow of a differential equation, is iterated upon as a function. Within this framework, there is a true underlying deterministic system which can be explained by a...

We will prove below that a system is inter-communicating if and only if it is chaotic. This is the fundamental law of data science, since it is both analytical, as coming from definitions, and synthetic, as explaining empirical observation, thus satisfying the Kantian transcendental deduction for...